In multiplication, the number 169 is famously recognized as the product of 13 multiplied by itself.
This means that 13 × 13 = 169. The number 169 is a perfect square, specifically the square of 13. This is a key property of 169, which arises from the multiplication of two identical integers. Interestingly, 13 itself is an odd prime number, meaning it is a whole number greater than 1 that cannot be formed by multiplying two smaller positive integers, and it is not divisible by 2.
Understanding Perfect Squares
A perfect square is an integer that can be expressed as the product of two equal integers. In other words, it is the square of an integer. The concept is fundamental in various areas of mathematics, from algebra to geometry.
- Definition: A number n is a perfect square if n = k × k for some integer k.
- Example: 9 is a perfect square because 3 × 3 = 9. Similarly, 169 is a perfect square because 13 × 13 = 169.
- Significance: Perfect squares are important in understanding number properties and simplifying square roots.
You can learn more about perfect squares on Wikipedia.
Factors of 169
The factors of a number are integers that divide into it without leaving a remainder. For 169, because it is a perfect square and 13 is a prime number, its factors are quite straightforward.
- Pair Factors: These are pairs of numbers that, when multiplied together, yield 169.
- Prime Factors: The prime numbers that multiply together to make 169.
The factors of 169 are 1, 13, and 169.
The prime factorization of 169 is 13 × 13. Since 13 is a prime number, this is its simplest form.
Pair Factors of 169
| Factor 1 | Factor 2 | Product |
|---|---|---|
| 1 | 169 | 169 |
| 13 | 13 | 169 |
This table clearly illustrates the multiplication pairs that result in 169. The unique characteristic of 169 is its single distinct pair of factors other than 1 and itself, which is 13 × 13. This reinforces its identity as a perfect square of an odd prime number. For further details on factors, you can explore resources like Math is Fun.
Practical Applications
Understanding perfect squares and factors like those of 169 has several practical applications:
- Geometry: Calculating the area of a square. If a square has an area of 169 square units, then each side must be 13 units long (since Area = side × side).
- Algebra: Solving equations involving squares, such as x² = 169, where the solution for x would be 13 (and -13).
- Simplifying Square Roots: When simplifying expressions like √169, knowing it's a perfect square allows for a direct answer of 13.
- Number Theory: Analyzing the divisibility rules and properties of numbers based on their prime factors.
Knowing that 169 equals 13 multiplied by 13 is a fundamental piece of multiplication knowledge that highlights the concept of perfect squares and the properties of prime numbers.
